Integral with Product of Exponentials of Cosines (Modified Bessel Function?)

Question

I am trying to analytically integrate the following, $$\int_{-\pi}^\pi \exp[a \cos(x - b)] \exp[c \cos(x - z)]dx,$$where $a$, $b$, $c$, and $z$ are all real constants. After seeing that the modified Bessel function of the first kind with order zero, $I_0$, satisfies the following, $\int_{-\pi}^\pi \exp[a \cos(x)]dx = 2\pi I_0(a)$, I thought it was plausible there was a fact about modified Bessel functions of the first kind with order zero I could exploit - but I haven't had any luck in finding similar integrals.